Hi Regina!
I'm a fan of both system fonts and web fonts. It's definitely okay for web sites to use system fonts. Combining web fonts and system fonts in a project can be a smart solution. It depends a lot on the project itself of course. Some projects really need high end web fonts to do their thing, and others can get away without them.
For most client projects I'll use either Typekit or Cloud Typography manage my web fonts. For demos and tutorials I tend to use Google Fonts (which has gotten a heck of a lot better than it used to be!). No matter which one you use, always give your font stack a little extra attention. Having more than just your web font and "serif" or "sans-serif" will make your typography stronger across all the various devices and contexts.
p.s. My two favourite system fonts are Georgia and Verdana. They're some amazingly hard working and durable fonts.